The present invention relates to fiber optic gyroscopes used for rotation sensing and, more particularly, to resonator fiber optic gyroscopes.
Fiber optic gyroscopes are an attractive means with which to sense rotation. They can be made quite small and still be constructed to withstand considerable mechanical shock, temperature change, and other environmental extremes. In the absence of moving parts, they can be nearly maintenance free, and they have the potential to become economical in cost. They can also be sensitive to low rotation rates that can be a problem in other kinds of optical gyroscopes.
There are various forms of optical inertial rotation sensors which use the well known Sagnac effect to detect rotation about a pertinent axis thereof. These include active optical gyroscopes having the gain medium contained in an optical cavity therein, such as the ring laser gyroscope, and passive optical gyroscopes without any gain medium in the primary optical path, such as the interferometric fiber optic gyroscope and the ring resonator fiber optic gyroscope. The avoidance of having the active medium along the primary optical path in the gyroscope eliminates some problems which are encountered in active gyroscopes such as low rotation rate lock-in, bias drift and some causes of scale factor variation.
Interferometric fiber optic gyroscopes typically employ a single spatial mode optical fiber of a substantial length formed into a coil, this substantial length of optical fiber being relatively costly. Resonator fiber optic gyroscopes, on the other hand, are constructed with relatively few turns of a single spatial mode optical fiber giving them the potential of being more economical than interferometric fiber optic gyroscopes. A resonator fiber optic gyroscope typically has three to fifty meters of optical fiber in its coil versus 100 to 2,000 meters of optical fiber in coils used in interferometric fiber optic gyroscopes. In addition, resonator fiber optic gyroscopes appear to have certain advantages in scale factor linearity and dynamic range.
In either type of passive gyroscope, these coils are part of a substantially closed optical path in which an electromagnetic wave, or light wave, is introduced and split into a pair of such waves, to propagate in opposite directions through the optical fiber coil to both ultimately impinge on a photodetector or photodetectors, a single photodetector for both waves in interferometric fiber optic gyroscopes and on corresponding ones of a pair of photodetectors inn resonator fiber optic gyroscopes. Rotation about the sensing axis of the core of the coiled optical fiber in either direction provides an effective optical path length increase in one rotational direction and an effective optical path length decrease in the opposite rotational direction for one member of this pair of electromagnetic waves. The opposite result occurs for the remaining member of the pair of electromagnetic waves for such rotation. Such path length differences between the pair of electromagnetic waves introduce corresponding phase shifts between those waves in interferometric fiber opticgyroscopes, or corresponding different optical cavity effective optical path lengths for these waves in a resonator fiber optic gyroscope.
In this latter instance, one or more optical frequency shifters are used to each effectively adjust the frequency of a corresponding one of the pair of electromagnetic waves that circulate in opposite directions in the resonator fiber optic coil. This is accomplished through such a frequency shifter shifting the frequency of a corresponding input electromagnetic wave giving rise to the resonator electromagnetic wave of interest. As a result, through feedback arrangements, the frequencies of each member of the pair of electromagnetic waves can be kept in resonance with the effective optical path length that wave is experiencing in the resonator fiber optic coil. Hence, any frequency difference between these waves becomes a measure of the rotation rate experienced by the resonator fiber optic coil about the axis around which this coil has been positioned. In such resonances, each wave has the portions thereof that previously were introduced in resonator 10 and have not yet dissipated, and the portions thereof currently being introduced in resonator 10, at a frequency such that they are all in phase with one another so they additively combine to provide a peak in the intensity of that wave in that resonator over a local range of frequencies.
The difference in frequency between the members of the pair of opposing electromagnetic waves in a resonant fiber optic gyroscope is desired to be constant when rotation conditions about the resonator optic fiber coil axis are unchanging thereby requiring that stable resonance conditions occur in that resonator in those circumstances. Furthermore, there are several advantages in achieving frequency shifting of the resonator electromagnetic waves by operating one or more integrated optics phase modulators for this purpose through each of which the corresponding input electromagnetic wave transmitted. These advantages involve economics, packaging volume, and performance. Obtaining a constant frequency difference between these resonator wave pair members using such a phase modulator requires that the phase modulator change phase in the form of a linear ramp since the derivative of phase with respect to time yields the frequency.
Because of the impossibility of having a phase modulator provide an infinite duration linear ramp with respect to time, a repetitive linear ramp with periodic resetting of the phase to a reference value must be used. The resulting sawtooth phase change waveform results in what is termed serrodyne phase modulation of those electromagnetic waves passing through the modulator.
Consider the known resonator fiber optical gyroscope system of FIG. 1. An optical cavity resonator, 10, formed by a continual path optical fiber is provided with an input directional coupler, 11, and an output directional optical coupler, 12. Resonator 10 is formed of a single spatial mode optical fiber which has two polarization eigenstates. Avoiding different optical path lengths for electromagnetic waves in each state is solved by thoroughly mixing the polarized waves in each state or, alternatively, permitting only one polarization eigenstate to exist by use of a polarizer. Such mixing is achieved by fabricating the resonator coil with two ends of a three to fifty meter length of such fiber spliced together so that the birefringence principal axes of the fiber are rotated 90.degree. with respect to each other on opposite sides of the splice, 13. Alternatively, instead of a splice, block 13 can represent a polarizer. The resonator fiber is characterized by a loss coefficient, .alpha., and, if a splice is used, an average of the propagation constants for the principal birefringence axes, .beta..sub.o, assuming an ideal 90.degree. splice. If a polarizer is used, the propagation constant will be that of the optical path of the permitted eigenstate of the electromagnetic waves which includes the transmission axis of the polarizer assuming a sufficiently large extinction ratio characterizes its blocking axis.
Directional coupler 11 is fabricated by appropriately fusing together an input optical fiber, 14, with the optical fiber in resonator 10, the fibers being tapered as they come into the fused portion on either side of that portion. Directional coupler provides a phase shift of .pi./2 between an input electromagnetic wave and the resulting electromagnetic wave at the resonator output thereof, the output wave further being characterized with respect to the input electromagnetic wave by a coupler coupling coefficient, k.sub.1, and a coupler loss coefficient, .gamma..sub.1. Directional coupler 11 has a suitable packaging arrangement thereabout.
Directional coupler 12 is constructed in generally the same manner as is directional coupler 11, but here an output optical fiber, 15, is fused to the optical fiber of resonator 10. Directional coupler 12 is characterized by a coupler coupling coefficient, k.sub.2, and a coupler loss coefficient, .gamma..sub.2.
The opposite ends of input optical fiber 14 are each connected to an optical integrated circuit, 16, formed with lithium niobate (LiNbO.sub.3) as the base material therefor. These ends of fiber 14 are appropriately coupled to integrated optical waveguides, 17 and 18, formed in the base material of optical integrated circuit 16. The relationship of the ends of input optical fiber 14 and the ends of integrated waveguides 17 and 18 are such that electromagnetic waves can be efficiently passed therebetween without undue losses. Integrated waveguide 17 is provided between a pair of metal plates formed on the base material of optical integrated circuit 16 to provide a phase modulator, 19, therein. Similarly, integrated waveguide 18 is formed between a another pair of metal plates formed on the base material to result in a further phase modulator, 20, in optical integrated circuit 16. Integrated waveguides 17 and 18 merge with one another into a single integrated waveguide, 21, to thereby provide a "Y" coupler in optical integrated circuit 16.
A laser, 22, is coupled to integrated waveguide 21 in a suitable manner so that light may be transmitted efficiently from laser 22 to integrated waveguide 21. Laser 22 is typically a solid state laser emitting electromagnetic radiation having a wavelength of 1.3 .mu.m with a spectral line width of one to hundreds of Khz. The wavelength at which laser 22 operates, or the frequency thereof, f.sub.o, can be adjusted by signals at an input thereof. Typical ways of providing such adjustment is to control the temperature of, or the current through, the solid state laser, or through the "pumping" semiconductor light emitting diode four the solid state laser, which in the latter instance may be a Nd:Yag laser. Where the diode is the emitting laser, the laser type may be an external cavity laser, a distributed feedback laser or other suitable types.
Thus, electromagnetic radiation emitted by laser 22 at a variable frequency f.sub.o is coupled to integrated waveguide 21, and from there split into two portions to form a pair of electromagnetic waves traveling in the input optical path in directions opposite one another. That is, the electromagnetic wave portion transmitted through integrated waveguide 17 proceeds therethrough and past phase modulator 19 into input optical fiber 14, and through input directional coupler 11 where a fraction k.sub.1 is continually coupled into resonator 10 to repeatedly travel therearound in a first direction, the counterclockwise directions there being a continual fractional loss for that wave of .gamma..sub.1 in coupler 11 as indicated above. The remaining portion of that wave, neither entering resonator 10 nor lost in coupler 11, continues to travel along input optical fiber 14 into integrated optical waveguide 18, through phase modulator 20, and finally through integrated waveguide 21 returning toward laser 22. Usually, laser 22 contains an isolator to prevent such returning waves from reaching the lasing portion thereof so that its properties are unaffected by those returning waves.
Similarly, the electromagnetic wave portion from laser 22, entering integrated waveguide 21 to begin in integrated waveguide 18, passes through phase modulator 20 into input optical fiber 14 and input directional coupler 11 where a fraction k.sub.1 thereof is continually coupled into resonator 10, accompanied by a continual fractional loss of .gamma..sub.1, to repeatedly traverse resonator 10 in a direction opposite (clockwise) to that traversed by the first portion coupled into resonator 10 described above. The remaining portion not coupled into resonator 10, and not lost in directional coupler 11, continues through input optical fiber 14 into integrated waveguide 17, passing through phase modulator 19, to again travel in integrated waveguide 21 in the opposite direction on its return to laser 22.
The pair of opposite direction traveling electromagnetic waves in resonator 10, a clockwise wave and a counterclockwise wave, each have a fraction k.sub.2 continually coupled into output optical fiber 15 with a fraction .gamma..sub.2 of each continually lost in coupler 12. The counterclockwise wave is transmitted by coupler 12 and fiber 15 to a corresponding photodetector, 23, and the clockwise wave is transmitted by them to a corresponding photodetector, 24, these photodetectors being positioned at opposite ends of output optical fiber 15. Photodetectors 23 and 24 are typically p-i-n photodiodes each of which is connected in corresponding one of a pair of bias and amplifying circuits, 25 and 26, respectively.
The frequency of the electromagnetic radiation emitted by laser 22, after being split from its combined form in integrated waveguide 21 into separate portions in integrated waveguides 17 and 18, has a resulting portion thereof shifted from frequency f.sub.o to a corresponding resonance frequency by a serrodyne waveform applied to phase modulator 19. The portion of the electromagnetic wave diverted into integrated waveguide 17 is shifted from frequency f.sub.o to frequency f.sub.o +f.sub.1 by phase modulator 19, and this frequency shifted electromagnetic wave is then coupled by input directional coupler 11 into resonator 10 as the counterclockwise electromagnetic wave. However, the portion of the electromagnetic wave directed into integrated waveguide 18 from integrated waveguide 21 is not shifted in frequency in the system of FIG. 1, although the frequency thereof could alternatively be similarly shifted from f.sub.o to f.sub.o +f.sub.2 by phase modulator 20 in forming the clockwise wave in coil 10. This arrangement would permit having to measure just differences in frequencies between the two serrodyne generators used in such an arrangement to obtain a system output signal rather than the absolute frequency value of a single generator which may be more convenient in some circumstances. The shifting of frequency of the wave in integrated waveguide 17 is caused by a serrodyne waveform applied to phase modulator 19 as indicated above, the serrodyne waveform for phase modulator 19 being supplied from a controlled serrodyne generator, 27. A similar serrodyne waveform would be applied to modulator 20 by a fixed frequency serrodyne generator if the wave in waveguide 18 was chosen to also be shifted in frequency.
Thus, controlled serrodyne generator 27 provides a sawtooth waveform output signal having a repetitive linear ramp variable frequency f.sub.1, the frequency f.sub.1 of this sawtooth waveform being controlled by an input shown on the upper side of generator 27 in FIG. 1. The repetitive linear ramp frequency of a sawtooth waveform from another serrodyne generator, if chosen as part of the control for modulator 20, would be fixed as indicated above, and held at a constant value, f.sub.2.
Structural detail of controlled serrodyne generator 27 is shown within the dashed line box representing that generator in FIG. 1 as three further blocks. The frequency control input of generator 27 is the input of a voltage-to-frequency converter, 27'. The frequency of the output signal of converter 27', proportional to the voltage at its input, sets the rate of count accumulation in a counter, 27", to which the output of converter 27' is connected. The output count totals of counter 27" are provided to a digital-to-analog converter, 27"', to form a "staircase" waveform to approximate the linear "ramps" occurring in a true serrodyne waveform.
The clockwise electromagnetic wave in resonator 10 and the counterclockwise electromagnetic wave in resonator 10 must always have the frequencies thereof driven toward values causing these waves to be in resonance in resonator 10 for the effective optical path length each is experiencing. This includes the path length variation resulting from any rotation of resonator 10 about the symmetrical axis thereof that is substantially perpendicular to the plane of the loop forming that optical resonator. Since controlled serrodyne generator 27 has the frequency of its serrodyne waveform controlled externally, that frequency value can be adjusted to the point that the corresponding counterclockwise wave in resonator 10 is in resonance with its effective path length, at least in a steady state situation. There, of course, can be transient effects not reflecting resonance in situations of sufficiently rapid changes of rotation rates of resonator 10.
On the other hand, the absence of a sawtooth waveform from another serrodyne generator to form part of the control of modulator 20 as shown in FIG. 1, or the use of a constant frequency for the sawtooth waveform of another serrodyne generator alternatively chosen to form part of the control of modulator 20, requires that the clockwise electromagnetic wave in resonator 10 be adjusted by other means. The means chosen in FIG. 1 is adjusting the frequency value of the light in laser 22. Thus, the adjustment of the value of the frequency f.sub.1 of the sawtooth waveform of controlled serrodyne generator 27 can be accomplished independently of the adjustment of the frequency f.sub.o of laser 22 so that, in steady state situations, both the counterclockwise electromagnetic wave and the clockwise electromagnetic wave in resonator 10 can be in resonance therein despite each experiencing a different effective optical path length therein.
Adjusting the frequency of the counterclockwise and clockwise electromagnetic waves traveling in opposite directions in resonator 10 means adjusting the frequency of each of these waves so that they are operating at the center of one of the peaks in the corresponding intensity spectra for resonator 10 experienced by such waves. Maintaining the frequency of the counterclockwise and the clockwise waves at the center of a corresponding resonance peak in the corresponding one of the resonator intensity spectra would be a difficult matter if that peak had to be estimated directly without providing some additional indicator of just where the center of the resonance peak actually is. Thus, the system of FIG. 1 introduces a bias modulation with respect to each of the counterclockwise and clockwise waves in resonator 10 through phase modulators 19 and 20, respectively. Such a bias modulation of each of these waves is used in a corresponding feedback loop to provide a loop discriminant characteristic followed by a signal therein which is acted on by that loop to adjust frequency f.sub.o and f.sub.1 as necessary to maintain resonance of the clockwise and counterclockwise waves, respectively.
A bias modulation generator, 28, provides a sinusoidal signal at a frequency f.sub.m to directly control modulator 20. Similarly, a further bias modulation generator, 29, provides a sinusoidal waveform of a frequency f.sub.n which is added to the sawtooth waveform at frequency f.sub.l provided by serrodyne generator 27. Frequencies f.sub.m and f.sub.n differ from one another to reduce the effects of electromagnetic wave backscattering in the optical fiber of resonator 10. The sinusoidal signal provided by bias modulation generator 28 is supplied to a node, 30. The addition of the sinusoidal signal provided by bias modulator generator 29 to the sawtooth waveform provided by serrodyne generator 27 is accomplished in a further summer, 31.
The sinusoidal waveform provided at node 30 is amplified in a power amplifier, 32, which is used to provide sufficient voltage to operate phase modulator 20. Similarly, the combined output signal provided by summer 31 is provided to the input of a further power amplifier, 33, used to provide sufficient voltage to operate phase modulator 19.
In this arrangement, the input electromagnetic wave to resonator 10 from integrated waveguide 17 will have an instantaneous electric field frequency of: EQU f.sub.o +f.sub.l -f.sub.n .DELTA..phi.sin.omega..sub.n t
The fraction of the electromagnetic wave reaching photodetector 23 through resonator 10 is not only shifted in frequency to a value of f.sub.o +f.sub.1, but is also effectively frequency modulated at f.sub.n. Depending on the difference between the resonance frequency and f.sub.o +f.sub.1, the intensity at that photodetector will thus have variations occurring therein at integer multiples of f.sub.n (though the fundamental and odd harmonics thereof will not occur at exact resonance). These latter components have amplitude factors related to the deviation occurring in the sum of (a) the phase shift resulting from the propagation constant multiplied by the path length in the counterclockwise direction in resonator 10, plus (b) phase shifts due to rotation and other sources, from a value equaling an integer multiple of 2.pi., a condition necessary for resonance along the effective optical path length in this direction.
The electromagnetic wave in integrated waveguide 18 enroute to resonator 10 will have instantaneous frequency equal to: EQU f.sub.o -f.sub.m .DELTA..phi.sin.omega..sub.m t
The fraction thereof reaching photodetector 24 through resonator 10 is at a frequency value in this instance of f.sub.o and frequency modulated at f.sub.m. Again, the intensity at photodetector 24 will have variations therein at integer multiples of f.sub.m, though not at the fundamental and odd harmonics thereof if these clockwise waves are at exact resonance. These latter components also have amplitude factors related to the deviation of the sum of (a) the phase shift resulting from the propagation constant multiplied by the path length in the clockwise direction in resonator 10, plus (b) phase shifts due to rotation and other sources, from a value equaling an integer multiple of 2.pi., again, a condition necessary for resonance along the effective optical path length in that direction.
Since the output signal of photodetector 24 has a frequency component at f.sub.m that is a measure of the deviation from resonance in resonator 10 in the clockwise direction, the output signal of bias and amplifier photodetector circuit 24 is provided to a filter, 34, capable of passing signal portions having a frequency component f.sub.m. Similarly, the output signal of photodetector 23 has a frequency component at f.sub.n that is a measure of the deviation from resonance in the counterclockwise direction, and so a filter, 35, is provided at the output of photodetector bias and amplifier circuit 25 capable of passing signal components having a frequency of f.sub.n.
The output signal from filter 34 is then provided to a phase detector, 36, at an operating signal input thereof. Phase detector 36 is a phase sensitive detector which also receives, at a demodulation signal input thereof, the output signal of bias modulation generator 28 which is the sinusoidal signal at frequency f.sub.m. Similarly, the output signal from filter 35 is provided to an operating signal input of a further phase detector, 37, which also receives at a demodulation input thereof the output sinusoidal signal at frequency f.sub.n of bias modulation generator 29. The output signals of phase detectors 36 and 37 follow a loop discriminant characteristic so that they indicate how far from resonance are the corresponding frequencies in resonator 10.
The discriminant characteristic followed by the output of phase detectors 36 and 37 will change algebraic sign for the frequencies on either side of the resonance peak and will have a zero magnitude at the resonance peak or resonance center. In fact, for sufficiently small values of the bias modulation generator output signals, the characteristic followed by the output signals of phase detectors 36 and 37 will be close to the derivative with respect to frequency of the intensity spectrum near the corresponding resonance peak. Thus, the output characteristics followed by the output signals of phase detectors 36 and 37 provide signals well suited for a feedback loop used to adjust frequencies to keep the corresponding electromagnetic waves in resonance in resonator 10.
Errors in the feedback loop are to be eliminated, and so the output signal of phase detector 36 is supplied to an integrator, 38, and the output signal of phase detector 37 is supplied to a further integrator, 39. Deviations from resonance are stored in these integrators which are then used in the loop to force the waves back to resonance in resonator 10. The output signal of integrator 38, in turn, is supplied to an amplifier, 40, used to provide signals to laser 22 to control the frequency f.sub.o of light being emitted by laser 22, thereby closing the feedback loop for adjusting that frequency. Similarly, the output signal of integrator 39 is supplied to an amplifier, 41, which in turn has its outputs supplied to the modulation input of controlled serrodyne generator 27, thus completing the remaining feedback loop to be used for adjusting serrodyne frequency f.sub.1.
However, certain errors can arise because of the effects of the propagation characteristics of resonator 10 on the electromagnetic waves oppositely propagating therein which lead to frequency differences therebetween that appear as though they were induced by rotations of resonator 10 about its axis of symmetry perpendicular to the plane in which it is positioned. One source of such error is the nonlinear behavior of the optical fiber material (primarily fused silica glass) in which these electromagnetic waves propagate resulting in differing indices of refraction being experienced by those waves in propagating through resonator 10.
The structure of the fused silica glass in the optical fiber used in resonator coil 10 has been found to give rise to a nonlinear polarization density that can be characterized as being of third order in the electric field. This means the material has a nonlinear dielectric tensor and so nonlinear indices of refraction which can differ for electromagnetic waves propagating in opposite directions through the coil. Thus, the propagation "constants" for the electromagnetic waves propagating in the clockwise and counterclockwise directions through coil 10 will exhibit an added nonlinear term depending on the intensities of the electric fields of the waves traveling therethrough, i.e. the optical Kerr effect. These added terms have been found to be expressible as: ##EQU1## where .DELTA..beta..sub.Kcw (t,z) is the change in propagation "constant" for the clockwise electromagnetic wave in resonator coil 10, and .DELTA..beta..sub.Kccw (t,z) is the change in the propagation "constant" for the counterclockwise traveling electromagnetic wave as a function of the distance travelled through coil 10 represented by z. The intensity I.sub.cw (t,z) is the intensity at a time t and position z along coil 10 of the clockwise wave, and the intensity I.sub.ccw (t,z) is the similar intensity for the counterclockwise wave traveling along coil 10. The Kerr coefficient is n.sub.2, and A represents the area of a cross section of the fiber in which the electromagnetic waves propagating therethrough are concentrated, with c being the speed of light in a vacuum.
As can be seen, the values of these last two expressions is different if I.sub.cw .noteq.I.sub.ccw indicating that differences in these added propagation "constant" terms can occur only upon differences occurring in the intensities of the clockwise and counterclockwise waves propagating in coil 10. Such differences in intensity are difficult if not impossible to avoid in practice, and so different propagation constants will be experienced by each of the counter-propagating waves in that coil, a situation which has been found to lead to corresponding resonant frequency differences between these waves which do not differ in nature from the resonant frequency differences arising from rotations of this coil. Hence, such nonlinear material behavior leads to errors in the output of the system of FIG. 1.
The nature of such errors arising because of the occurrence of these nonlinear terms in the propagation "constants" for electromagnetic waves in coil 10 can be found using a suitable representation for these waves propagating in coil 10. One such representation that can be shown to be suitable for the clockwise wave is given as follows: ##EQU2## where z has a value of zero at the output of coupler 11 for clockwise waves, a value of 11 at the input to coupler 12 for clockwise waves, and a value of L at the input to coupler 11 for clockwise waves with the couplers assumed to have no significant extent along the z path. Thus, the distance from coupler 11 to coupler 12 not passing through splice (or polarizer) 13 is l.sub.1, and the distance from coupler 12 to coupler 11 through splice (or polarizer) 13 is l.sub.2 with L=l.sub.1+l.sub.2.
The effective propagation "constant" in the foregoing equation, .beta..sub.cw, gives the effective phase change per unit length along coil 10, and comprises a pair of terms, that is .beta..sub.cw =.beta..sub.o -.DELTA..beta..sub.m sin.omega..sub.m t. The term .beta..sub.o =2.pi.n.sub.eff f.sub.o /c is the weighted average of the propagation constants of the two principle axes of birefringence of the optical fiber in resonator 10 if a splice 13 has been used. This average is based on the fraction of travel over each axis by the electromagnetic waves in the resonator in the corresponding polarization state with changes between axes being due to the 90.degree. rotation splice in the optical fiber of that resonator as described above. A rotation of other than 90.degree. will give an uneven weighting to these axes. If, on the other hand, a polarizer is used rather than a splice at block 13, there will be only single propagation constants as n.sub.eff will no longer be an average of indices of refraction but a single value index refraction (ignoring other index of refraction issues). Again, the parameter .theta. in the above equations for E.sub.cw reflects any added phase due to the 90.degree. splice, or near 90.degree. splice, involving block 13 if present rather than a polarizer.
The parameter .DELTA..beta.=2.pi.n.sub.eff f.sub.m .DELTA..phi..sub.m /c is the equivalent change in the effective propagation constant due to the incoming electromagnetic waves having been modulated sinusoidally at the rate .omega..sub.m with a peak amplitude change of .DELTA..phi..sub.m. The parameter .+-..phi..sub.r represent the Sagnac phase shift induced by rotation in one direction or another about the axis symmetry of resonator 10 perpendicular to a plane passing through all of that resonator. The coefficient .alpha. is the coefficient giving the loss per unit length in the resonator optical fiber of coil 10. The factor q represents the division of the electromagnetic wave from laser 22, E.sub.in, due to the splitting of that wave by "y" coupler 21 and also the losses for that wave accumulated on the way to input directional coupler 11. Of course, .omega..sub.o =2.pi.f.sub.o, or the frequency of oscillation in the electromagnetic wave provided by laser 22. The parameter u is the counting parameter of the number of circulations about coil 10 by the electromagnetic waves. Finally, the parameter .theta..sub.Kcw represents the phase change for one round trip through coil 10 in the electromagnetic waves in the clockwise direction due to the Kerr effect.
Although the last equation is indeed just for the clockwise traveling electromagnetic wave in resonator 10 that began in integrated optical waveguide 18, the counterpart equation for waves beginning in integrated waveguide 17 and traveling in the opposite or counterclockwise direction in resonator 10 will be quite similar and so is not separately set forth here. Such counterclockwise waves will, however, have the opposite sign for any rotation induced phase shift and will have a slightly different propagation "constant" .beta. because of the frequency shifts due to the use of serrodyne generator 27. Thus, .beta..sub.ccw =.beta..sub.o-1 -.DELTA..beta..sub.n sin.theta..sub.n t. Then, .beta..sub.o-1 =2.pi.n.sub.eff (f.sub.o +f.sub.1)/c and .DELTA..beta..sub.n =2.pi.n.sub.eff f.sub.n .DELTA..phi..sub.n /c where .DELTA..phi..sub.n is the peak amplitude of the bias modulation sinusoid.
From the foregoing equation for E.sub.cw, and from the counterpart equation for E.sub.ccw not set out here, the intensities associated with these propagating electromagnetic waves, I.sub.cw (t, z) and I.sub.ccw (t, z) can be found. Thus, ##EQU3## Similarly, ##EQU4## The well known limit for the infinite geometric series has been used in obtaining these equations for the clockwise intensity as has the well known Euler equation.
In a similar manner, the counterclockwise intensity is found ##EQU5## where p represents the split of E.sub.in going into waveguide 17 as well as the losses accumulated propagating to directional input coupler 11, and where EQU .DELTA..sub.ccw =.beta..sub.0-1 L-.DELTA..beta..sub.n Lsin.omega..sub.n t-.phi..sub.r +.theta.=.beta..sub.ccw L-.phi..sub.r +.theta.
Here, .theta..sub.Kccw represents the phase change in resonator coil 10 for one passage of the counterclockwise electromagnetic wave therethrough due to the optical Kerr effect. Also, ##EQU6## These equations can be further consolidated by appropriate substitutions based on the following definition which will be made further along in this text: ##EQU7## making use of a trigonometric identity.
These intensity equations can then be used to evaluate .theta..sub.Kcw since the clockwise Kerr effect phase error can be found by integrating the change in the propagation "constants" due to the Kerr effect over the optical path through resonator coil 10, or ##EQU8## From the equation above for .DELTA..beta..sub.Kcw (t, z), this last expression can be rewritten as ##EQU9## The integrals involved in this last expression can be evaluated and shown to be ##EQU10## These expressions for the integrals can be simplified by introducing l.sub.1 =L/2, which is typically the situation occurring in the system of FIG. 1 although not a required condition for successful operation of the system. If that introduction is made, the terms in the brackets in the evaluations of the above integrals will be equal allowing the expression for the .theta..sub.Kcw to be written as: EQU .theta..sub.Kcw =.delta.I.sub.o [q.sup.2 .GAMMA.(.DELTA..sub.cw +.theta..sub.Kcw)+2p.sup.2 .GAMMA.(.DELTA..sub.ccw +.theta..sub.Kccw)] EQU where EQU I.sub.o =E.sub.in.sup.2
and ##EQU11## In a similar manner, .theta..sub.Kccw can be found to be EQU .theta..sub.Kccw =.delta.I.sub.0 [p.sup.2 .GAMMA.(.DELTA..sub.ccw +.theta..sub.Kccw)+2q.sup.2 .GAMMA.(.DELTA..sub.cw +.theta..sub.Kcw)]
As indicated above, the clockwise traveling electromagnetic wave portion reaching photodiode 24, I.sub.cw-d, will have the frequency thereof controlled in the feedback loop operating laser 22 to set the value f.sub.o to keep that electromagnetic wave in resonance in resonator coil 10 in steady state conditions. This is accomplished in the feedback loop for laser 22 by forcing any bias modulation frequency component at the bias modulation frequency .omega..sub.m in I.sub.cw-d to zero through shifting the value of f.sub.o sufficiently for the clockwise wave to be in resonance. Such feedback action yields a clockwise wave intensity at photodiode 24 of ##EQU12## In such a resonance condition, the total phase change of the clockwise wave over the optical path through the resonator optical fiber coil 10, .DELTA..sub.cw +.theta..sub.Kcw, must equal an integral number of cycles to be able to stably reproduce itself over that path. The parameter .theta..sub.Kcw is the time average value of the Kerr effect phase change .theta..sub.Kcw.
This resonance condition can be expressed as .beta..sub.o L.+-..phi..sub.r +.theta..sub.Kcw =2m.pi. assuming use of polarizer for block 13 (otherwise the splice angle .theta. must be included).
In a similar manner, the portion of the counterclockwise wave in resonator coil 10 impinges on photodiode 23, and the feedback loop beginning therefrom to control serrodyne generator 27 adjusts the frequency of the counterclockwise wave, .omega..sub.o +.omega..sub.1, to keep that wave in resonance in resonator optical fiber coil 10 in steady state conditions. Again, this is accomplished in this feedback loop through forcing to zero any bias modulation signal frequency component at bias modulation frequency .omega..sub.n in the counterclockwise wave intensity I.sub.ccw-d on photodiode 23 giving the result ##EQU13## In these circumstances at the resonance condition, again the phase change in the counterclockwise direction in the optical path in coil 10, .DELTA..sub.ccw +.theta..sub.Kccw, for the counterclockwise electromagnetic wave must be an integral number of cycles for stable reproduction of that wave over that path. This condition can be expressed as .beta..sub.o-1 L.+-..phi..sub.r +.theta..sub.Kccw =2m.pi., assuming use of a polarizer for block 13, where again m is an integer and .theta..sub.Kccw is the time average value of the Kerr effect phase change over that optical path.
These two resonance conditions in the preceding paragraphs are, as indicated, maintained during steady state conditions in the system of FIG. 1. Thus, any changes in any of the terms in these resonance condition equations must balance one another if those conditions are to be maintained. As a result, the following conditions must also hold: EQU .DELTA..sub.cw +.theta..sub.Kcw =o; .DELTA..sub.ccw +.theta..sub.Kccw =o
The bar over the top of the symbols used again denotes that the time average value is taken. The effects in .phi..sub.r, .theta..sub.Kcw and .theta..sub.Kccw of the harmonics of the modulation frequencies in the signals at photodiodes 23 and 24 are so much smaller than both 2.pi. and the amplitudes of the bias modulations .DELTA..phi..sub.m and .DELTA..phi..sub.n that they may be ignored.
Using these last two equations, and the expressions found above for .theta..sub.Kcw and .theta..sub.Kccw, gives the result: EQU .DELTA..sub.cw =-.delta.I.sub.o [q.sup.2 .GAMMA.(.DELTA..beta..sub.m Lsin.omega..sub.m t)+2p.sup.2 .GAMMA.(.DELTA..beta..sub.n Lsin.omega..sub.n t)] EQU .DELTA..sub.ccw =-.delta.I.sub.o [p.sup.2 .GAMMA.(.DELTA..beta..sub.n Lsin.omega..sub.n t)+2q.sup.2 .GAMMA.(.DELTA..beta..sub.m Lsin.omega..sub.m t)]
Thus, under the bias modulations over the resonances at bias modulation frequencies .omega..sub.m and .omega..sub.n, the time average change in phase in the clockwise direction from resonance set by the bias modulation feedback loops, .DELTA..sub.cw, equals the time average change in phase over the optical path due to the presence of the optical Kerr effect. This time average phase change due to the Kerr effect occurs because the bias modulation signal amplitudes affect the time average intensity in the resonator differently in each propagation direction in resonator 10. This result, in turn, causes the feedback loops to maintain optical frequencies that are not due to the rotation rate of resonator 10 alone, but that are also due to the presence of the optical Kerr effect thus leading to error. This situation is also true of the time average change in phase from resonance in the counterclockwise direction, .DELTA..sub.ccw, set by these loops. These last two expressions would otherwise be zero in the absence of the Kerr effect.
As is well known, the frequency difference between the clockwise and counterclockwise electromagnetic waves in resonator coil 10 for a rotation rate .OMEGA. is given by: ##EQU14## where A is the area enclosed by resonator coil 10, P is the perimeter of that area, and .lambda. is the wavelength of the center of the spectrum of the electromagnetic waves emitted by source 11. Thus, the effective rotation rate error due to the Kerr effect, .OMEGA..sub.Ke, can be written: ##EQU15## where .DELTA.f.sub.Ke here is the resonant frequency difference between the clockwise and counterclockwise waves due to the optical Kerr effect. This frequency difference is equal to the total phase difference which occurs between the clockwise and counterclockwise waves around the perimeter of ring 10 because of the Kerr effect, .DELTA..sub.cw -.DELTA..sub.ccw, divided by 2.pi. giving the number of amplitude wave cycles over this perimeter, divided by the time of propagation over this perimeter n.sub.eff P/c, i.e. .DELTA.f.sub.Ke =(.DELTA..sub.cw -.DELTA..sub.ccw /2.pi.)c/n.sub.eff P.
Thus, the expression above for the rotation rate error due to the Kerr effect .OMEGA..sub.Ke can be rewritten using the expression therefor given above, and this last expression along with the above expressions for .DELTA..sub.cw and .DELTA..sub.ccw to give the following result: ##EQU16## Therefore, the rotation rate error .OMEGA..sub.Ke can be evaluated by finding values for the two time averages occurring therein, or ##EQU17## The integrals in these last expressions have been evaluated using a small angle approximation based on the amplitude of the phase changes due to the bias modulations being relatively small. If it is also assumed, as is typical, that the difference between the phase change amplitudes of the bias modulations are small, or EQU .DELTA..beta..sub.m .apprxeq..DELTA..beta..sub.n
and the following definition is made ##EQU18## In these circumstances, the rotational error rate due to the optical Kerr effect becomes: ##EQU19##
This result for the rotational rate error due to the Kerr effect can be written in simplified form by the introduction of two constants, c.sub.1 and c.sub.2, defined as follows: ##EQU20## Then the expression above for the rotational rate error due to the Kerr effect can be written as: EQU .OMEGA..sub.Ke =C.sub.1 I.sub.o {(q.sup.2 -p.sup.2)-C.sub.2 (.DELTA..beta..sub.m -.DELTA..beta..sub.n)L(p.sup.2 +q.sup.2)}.
Thus, the rotational rate error due to the Kerr effect can be seen to depend linearly on the input intensity of the electromagnetic waves provided by laser 22. In addition, the error can be seen to arise because of unequal counter-rotating electromagnetic waves due to unequal fractions p and q of the input waves being converted to such counter-rotating waves at integrated waveguide junction 21 providing a "Y" coupler in optical integrated circuit 16, compounded by any inequalities in the bias modulation amplitudes and frequencies resulting in unequal corresponding modulation propagation "constants" .DELTA..beta..sub.m and .DELTA..beta..sub.n. In practice, such inequalities are often, if not usually, unavoidable so that resultant Kerr effect errors are present.
To achieve suitable accuracy in a resonator fiber optic gyroscope in many applications, such errors must be substantially reduced or eliminated. One manner of correcting errors of this nature has been set out in the U.S. Pat. No. 4,673,293 to Sanders. There, feedback has been used to alter the intensity of one of the propagating electromagnetic waves to force an error signal based on errors of the foregoing nature towards the value of zero. However, this arrangement requires use of an intensity modulator in the path of one of the counter-propagating electromagnetic waves to effect control thereof. A modulator of that type of sufficient capability is typically expensive because of the difficulties in fabricating one suitable for the intended use. Therefore, there is a desire to provide a fiber optic gyroscope which overcomes the present errors due to the Kerr effect in an alternative manner.